Mechanic: Sorrifier-Driven Formal Decomposition Workflow for Automated Theorem Proving

TL;DR

Mechanic introduces a sorry-driven decomposition approach for automated theorem proving, effectively isolating and resolving subproblems independently to improve proof success rates and efficiency in complex mathematical reasoning.

Contribution

It presents a novel agent system that uses sorry placeholders for formal decomposition, enabling efficient handling of failed proofs without full regeneration or excessive context growth.

Findings

Achieves significant improvements on IMO 2025 and Putnam 2025 benchmarks.

Reduces proof attempt failures and improves proof efficiency.

Effectively isolates unresolved subgoals for independent resolution.

Abstract

Recent advances in large language models (LLMs) and LLM-based agents have substantially improved the capabilities of automated theorem proving. However, for problems requiring complex mathematical reasoning, current systems rarely succeed on the first try and must repeatedly modify their proof strategies. Existing approaches for handling failed attempts typically either discard the entire proof and regenerate it from scratch or iteratively fix errors within the proof. The former is inefficient, as it may abandon mostly correct reasoning due to localized errors, while the latter, although preserving prior progress, leads to progressively longer contexts which progressively degrades the model's ability to attend to the remaining unresolved subproblems. To address this dilemma, we propose Mechanic, a novel agent system that employs a sorry-driven formal decomposition strategy. By leveraging the sorry placeholder in Lean to precisely isolate unresolved subgoals while preserving the surrounding verified proof structure, Mechanic extracts each failed subproblem into a clean, self-contained context and resolves it independently. This avoids both the waste of full regeneration and the excessive context length induced by repeated repairs. Experimental results on challenging mathematical competition benchmarks, including IMO 2025 and Putnam 2025, demonstrate that our agent achieves significant advantages in proving efficiency.